Question

which equation represents a line that is perpendicular to line mn?

\\(y = -\frac{1}{4}x + 9\\)
\\(y = \frac{1}{4}x + 6\\)
\\(y = -4x - 7\\)
\\(y = 4x - 6\\)

Explanation:

Identify coordinates of points M and N

Using the Linear Functions knowledge point, we locate points \(M\) and \(N\) on the coordinate plane. Note that the axes are oriented with the horizontal axis as \(x\) and the vertical axis as \(y\), but the image is rotated \(90^\circ\) clockwise. Let's read the coordinates carefully by aligning with the standard grid labels:

• Point \(M\) is located at \(x = 6\), \(y = 7\).
• Point \(N\) is located at \(x = 3\), \(y = -5\).

Calculate the slope of line MN

Using the Rate of Change knowledge point, we find the slope \(m_1\) of the line passing through \(M(6, 7)\) and \(N(3, -5)\):

$$ m_1 = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-5 - 7}{3 - 6} = \frac{-12}{-3} = 4 $$

Determine the perpendicular slope

Two lines are perpendicular if their slopes are negative reciprocals of each other:

$$ m_2 = -\frac{1}{m_1} = -\frac{1}{4} $$

Match with the given equations

Using the Slope-Intercept Form knowledge point, we look for an equation of the form \(y = mx + b\) where the slope \(m = -\frac{1}{4}\).
The given options are:

• \(y = -\frac{1}{4}x + 9\)
• \(y = \frac{1}{4}x + 6\)
• \(y = -4x - 7\)
• \(y = 4x - 6\)

The equation with a slope of \(-\frac{1}{4}\) is \(y = -\frac{1}{4}x + 9\).

Answer:

• (A) \(y = -\frac{1}{4}x + 9\) (Correct answer)
• (B) \(y = \frac{1}{4}x + 6\)
• (C) \(y = -4x - 7\)
• (D) \(y = 4x - 6\)